THE MOON'S MERIDIAN ALTITUDE. 279 



proximate time at Greenwich: for this time rind the rate or 

 change or' the moon's declination lor one minute of time, and 

 also the difference between the moon's meridian altitude, and 

 its altitude one minute before or after, caused by the diurnal 

 rotation of the earth, combined with the progress in right as- 

 cension; those two effects may be considered (at small distances 

 from the meridian) as operating in the same line, and in op- 

 posite directions. Put x, to represent the time from the meridi- 

 an, when the moon's altitude will be the same as when on the 

 meridian, a, for the change in declination, and b, for the de- 

 pression from the meridian in one minute of time; the first 

 increases or diminishes in the direct, and the second (for small 

 quantities) in the duplicate ratio of the times ; hence we shall 



have bx* = ax, and therefore x = 7; i. e. the time (in minutes) 



b ' 



from the meridian, when the change in declination will be 



equalized by the depression, will be found by dividing the 



change of declination by the depression for one minute ot 



time; at half this distance in time from the meridian, the change 



in declination will be double the depression, and will be the 



■ a * 

 maximum, or point of greatest altitude, therefore — - will re- 

 present the correction; i. e. the square of the rate of change 

 of declination, divided by four times the depression for one 

 minute of time, will be the correction for the moon's meridian 

 altitude; which may be conveniently found by the following 

 formula, expressed in logarithmic language. 



RULE. 



To twice the sine of 14-' 29" 5 add the arith. comp. of 

 the sine (or cosecant) of 1' and the logarithm of 120, the sum, 

 abating 20 from the index, will produce the constant logarithm 

 .86514-14'. To the constant logarithm add the cosine of the 

 declination of the moon, the cosine of the latitude, and the 

 arith. comp. of the cosine (or secant) of the altitude, the sum, 

 rejecting tens from the index, will be the logarithm of four 

 times the depression, which subtracted from twice the loga- 



o 



